#432366

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

911 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,15,19,24)(8,12)(9,13)(10,14)(11,18,25,17)(16,31,34,30)(20,27)(21,28)(22,29)(23,32,26,33)(35,36);(5,7,9,17,10,19,22,18)(6,11,13,24,14,25,29,15)(8,16,20,32,21,34,35,33)(12,26,27,30,28,23,36,31);(5,8,9,20,10,21,22,35)(6,12,13,27,14,28,29,36)(7,16,17,32,19,34,18,33)(11,23,24,31,25,26,15,30);(2,4)(7,17,19,18)(11,24,25,15)(16,32,34,33)(23,31,26,30)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=ef,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339562);